The only "health problem" that I've noticed comes from really long play sessions with very limited breaks which makes the muscles in my forearm start to cramp up a little bit.
If you keep playing through the pain and cramped forearm I suspect that you could cause some muscular damage (haven't tried this one).
The way to combat this (for me, at least) was to thoroughly stretch my forearms and my fingers daily and do a little bit of stretching before playing sessions and occasionally take short breaks and stretch out my forearms.
Learning to stretch properly really improved my osu!-experience as it fixed most of my previous forearm issues and dramatically increased the mobility of my wrist (can now use tablets areas twice as big as I could previously without any pain).
Previous IssuesI had some issues with tight and cramped forearm muscles even before starting with osu! from years in front of my PC. Even though all the stretching pretty much fixed my forearm it occasionally starts to cramp up (not as in a full on painful cramps but the muscle becomes tighter and range of motion becomes limited) from longer playing sessions. The more I play after the muscle starts to feel "tighter" the tighter it becomes until the muscle is painfully cramping, so I just stop as soon as I feel a muscle in my forearm starting to feel tense/tight. Then I stretch affected muscles and wait for them to "return to normal" before playing again (30 minutes-about a day depending on how soon I stop playing after the tightness feeling appears).
Not sure if osu! alone would cause these problems but it did make them a lot worse for me, so I would assume that enough "improper" playing would start cramping up forearms and hands even on someone without any preexisting issues.
I would recommend that most osu!-players who don't already know this to look up how to properly stretch the different muscles in your hands and forearms,
especially if you have either limited wrist mobility, tight forearm muscles or experience pain in your forearm/hand/wrist while playing.
And also, take frequent short breaks and just stretch out tight muscles while playing.
**I AM NOT A DOCTOR, THIS IS JUST WHAT WORKED FOR ME.**
*Sorry for wall of text
TL;DR:
Learn how to stretch properly, greatly improved my osu! for me.
